An introduction toFractal Geometry

ABBAS KARIMI

Complex Systems & Network Science Group (CSNS)

Shahid Beheshti University (SBU)

May 01, 2017

Sitpor.org

Introduction

“Unfortunately, the world has not been designed for the convenience of mathematicians.”

Benoît Mandelbrot

Born: 20 November 1924, Warsaw, Poland

Died: 14 October 2010 (aged 85), Cambridge, Massachusetts, United

States

Image: At a TED conference in 2010.@wikipedia

Introduction

length of the small intestine:

2.75 - 10.49 m!

But, HOW?Wikipedia

Self-SimilarityIf we walk the coastline then we discover even more fine detail. Structures which exhibit a similar pattern whatever the scale are said to be self-similar.

Self-Similarity

Fractional Dimension!How many little things fit inside a big thing?

ShapeM.

FactorNumber of small

copies

line 3 3

square 3 9

cube 3 27

SnowflakeD = 1.46

Seriously?!Why not D = 2 ?!

D(Sierpiński Triangle) > D(Snowflake) 1.58 > 1.46

Self-Similar Dimension, let’s call it ROUGHNESS!

https://www.ted.com/talks/benoit_mandelbrot_fractals_the_art_of_roughness

Self-Similarity VS Topological Dimension!A fractal is a geometrical object whose self-similarity dimension is greater than its topological dimension.

● The Sierpinski triangle has a self-similarity dimension of 1.585 and a topological dimension of 1.

● The Cantor set has a self-similarity dimension of 0.6309 and a topological dimension of 0!

Random Fractals1. Random Koch Curve2. Chaos Game3. Collage Theorem

Koch CurveA self-similar shape. A small copy of the curve, when magnified, exactly resembles the full curve. D=log4/log3. See Falconer, 2003, Chapter 15

Random Koch CurveA small copy of the curve, when magnified, closely resembles the full curve, but it is not an exact replica. (D=log4/log3) Statistical self-similarity!

Irregular Fractals are not Random!● We make them by using an asymmetric or irregular generation rule.● They are produced with a deterministic rule so they are NOT random!● Nevertheless, the resultant shape has a somewhat random or

disordered feel to it!

Sierpiński Carpet is not random!

Chaos GameThis is a stochastic (not deterministic) dynamical

system; an element of chance is incorporated in each

step!

100 points

1000 points

100,000 points

➔ The chaos game shows that a random dynamical system can give a deterministic result.

➔ #emergence_of_order

Collage Theorem

Given essentially any shape (not only fractals), one can make a chaos game that will generate it.

● Since in finding the particular affine transformations needed to reproduce an image, one forms a collage in which the full shape is covered with several smaller shapes. These smaller shapes then define the affine transformations that will generate the full shape.

● Application: Image compression

Collage Theorem

An image of a fern-like fractal that exhibits affine self-similarity. Each of the leaves of the fern is related to each other leaf by an affine transformation.The red leaf can be transformed into both the small dark blue leaf and the large light blue leaf by a combination of reflection, rotation, scaling, and translation.

Julia SetMandelbrot Set

Julia SetThe julia set for a function is simply the collection of all initial conditions that do not tend to infinity when iterated with that function.

❖ 2,4,16, … diverging ❖ -3, 9, 81, … diverging

❖ 0.5, 0.25, 0.0625, … ,0.0 converging❖ Filled Julia Set = {-1,1}

Julia Set, The complex squaring function

❖ Filled Julian Set is a circle with r = 1

Julia Set, What about this function?

Julia Set, What about this function?

 

Julian Set

Other functions?

Mandelbrot Set

The Mandelbrot set consist of the set of all parameter values c for which the julia set of “ ” is connected.

➔ Set of complex numbers c for which the function above does not diverge when iterated from z=0, i.e., for which the sequence remains bounded in absolute value:

c, c²+c, (c²+c)²+c, ((c²+c)²+c)²+c, (((c²+c)²+c)²+c)²+c, …

Mandelbrot Set

REFERENCES

Sitpor.org/series + wikipedia.org

Thank You :D

ABBAS KARIMI

Complex Systems & Network Science Group (CSNS)

Shahid Beheshti University (SBU)

May 01, 2017

Sitpor.org